Calculus of Variations – What Are Natural Boundary Conditions?

Question

What do people mean, when they speak of natural boundary conditions in the calculus of variations?

How do natural boundary conditions relate to the Euler-Lagrange equations?

An example would be fantastic!

Answer 1

Basically two types of boundary conditions are used: Essential or geometric boundary conditions which are imposed on the primary variable like displacements, and Natural or force boundary conditions which are imposed on the secondary variable like forces and tractions. Essential boundary conditions are imposed explicitly on the solution but natural boundary conditions are automatically satisfied after solution of the problem.

Natural Boundary Conditions of the Simplest Kind:

Let $~J : C^2[x_0, x_1] → \mathbb R~$ be a functional of the form of $$J(y) = \int^{x_1}_{x_0}f(x, y, y') dx$$ and assume that no boundary conditions have been imposed on $~y~$, then $~J~$ have an extremum $~y~$ if the following necessary conditions are satisfied:

$(i)~~$ The ordinate of the extremal satisfies the Euler-Lagrange Equation $$f_y - \frac{d}{dx} f_y' = 0$$

$(ii)~~$ At $~x = x_0~$ $$\left|\frac{\partial f}{\partial y'}\right|_{~x_0}=0$$

$(iii)~~$ At $~x = x_1~$ $$\left|\frac{\partial f}{\partial y'}\right|_{~x_1}=0$$